3.336 \(\int \frac{(A+B x) \left (a+c x^2\right )^{3/2}}{x^6} \, dx\)

Optimal. Leaf size=93 \[ -\frac{A \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{3 B c \sqrt{a+c x^2}}{8 x^2}-\frac{B \left (a+c x^2\right )^{3/2}}{4 x^4} \]

[Out]

(-3*B*c*Sqrt[a + c*x^2])/(8*x^2) - (B*(a + c*x^2)^(3/2))/(4*x^4) - (A*(a + c*x^2
)^(5/2))/(5*a*x^5) - (3*B*c^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(8*Sqrt[a])

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Rubi [A]  time = 0.152028, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{A \left (a+c x^2\right )^{5/2}}{5 a x^5}-\frac{3 B c^2 \tanh ^{-1}\left (\frac{\sqrt{a+c x^2}}{\sqrt{a}}\right )}{8 \sqrt{a}}-\frac{3 B c \sqrt{a+c x^2}}{8 x^2}-\frac{B \left (a+c x^2\right )^{3/2}}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*(a + c*x^2)^(3/2))/x^6,x]

[Out]

(-3*B*c*Sqrt[a + c*x^2])/(8*x^2) - (B*(a + c*x^2)^(3/2))/(4*x^4) - (A*(a + c*x^2
)^(5/2))/(5*a*x^5) - (3*B*c^2*ArcTanh[Sqrt[a + c*x^2]/Sqrt[a]])/(8*Sqrt[a])

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Rubi in Sympy [A]  time = 13.8673, size = 87, normalized size = 0.94 \[ - \frac{A \left (a + c x^{2}\right )^{\frac{5}{2}}}{5 a x^{5}} - \frac{3 B c \sqrt{a + c x^{2}}}{8 x^{2}} - \frac{B \left (a + c x^{2}\right )^{\frac{3}{2}}}{4 x^{4}} - \frac{3 B c^{2} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{2}}}{\sqrt{a}} \right )}}{8 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+a)**(3/2)/x**6,x)

[Out]

-A*(a + c*x**2)**(5/2)/(5*a*x**5) - 3*B*c*sqrt(a + c*x**2)/(8*x**2) - B*(a + c*x
**2)**(3/2)/(4*x**4) - 3*B*c**2*atanh(sqrt(a + c*x**2)/sqrt(a))/(8*sqrt(a))

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Mathematica [A]  time = 0.28257, size = 104, normalized size = 1.12 \[ -\frac{\frac{\sqrt{a+c x^2} \left (2 a^2 (4 A+5 B x)+a c x^2 (16 A+25 B x)+8 A c^2 x^4\right )}{x^5}+15 \sqrt{a} B c^2 \log \left (\sqrt{a} \sqrt{a+c x^2}+a\right )-15 \sqrt{a} B c^2 \log (x)}{40 a} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*(a + c*x^2)^(3/2))/x^6,x]

[Out]

-((Sqrt[a + c*x^2]*(8*A*c^2*x^4 + 2*a^2*(4*A + 5*B*x) + a*c*x^2*(16*A + 25*B*x))
)/x^5 - 15*Sqrt[a]*B*c^2*Log[x] + 15*Sqrt[a]*B*c^2*Log[a + Sqrt[a]*Sqrt[a + c*x^
2]])/(40*a)

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Maple [A]  time = 0.017, size = 125, normalized size = 1.3 \[ -{\frac{A}{5\,a{x}^{5}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{B}{4\,a{x}^{4}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{Bc}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{B{c}^{2}}{8\,{a}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,B{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{2}+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{3\,B{c}^{2}}{8\,a}\sqrt{c{x}^{2}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+a)^(3/2)/x^6,x)

[Out]

-1/5*A*(c*x^2+a)^(5/2)/a/x^5-1/4*B/a/x^4*(c*x^2+a)^(5/2)-1/8*B/a^2*c/x^2*(c*x^2+
a)^(5/2)+1/8*B/a^2*c^2*(c*x^2+a)^(3/2)-3/8*B/a^(1/2)*c^2*ln((2*a+2*a^(1/2)*(c*x^
2+a)^(1/2))/x)+3/8*B/a*c^2*(c*x^2+a)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^6,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.34895, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, B a c^{2} x^{5} \log \left (-\frac{{\left (c x^{2} + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{c x^{2} + a} a}{x^{2}}\right ) - 2 \,{\left (8 \, A c^{2} x^{4} + 25 \, B a c x^{3} + 16 \, A a c x^{2} + 10 \, B a^{2} x + 8 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{a}}{80 \, a^{\frac{3}{2}} x^{5}}, -\frac{15 \, B a c^{2} x^{5} \arctan \left (\frac{\sqrt{-a}}{\sqrt{c x^{2} + a}}\right ) +{\left (8 \, A c^{2} x^{4} + 25 \, B a c x^{3} + 16 \, A a c x^{2} + 10 \, B a^{2} x + 8 \, A a^{2}\right )} \sqrt{c x^{2} + a} \sqrt{-a}}{40 \, \sqrt{-a} a x^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^6,x, algorithm="fricas")

[Out]

[1/80*(15*B*a*c^2*x^5*log(-((c*x^2 + 2*a)*sqrt(a) - 2*sqrt(c*x^2 + a)*a)/x^2) -
2*(8*A*c^2*x^4 + 25*B*a*c*x^3 + 16*A*a*c*x^2 + 10*B*a^2*x + 8*A*a^2)*sqrt(c*x^2
+ a)*sqrt(a))/(a^(3/2)*x^5), -1/40*(15*B*a*c^2*x^5*arctan(sqrt(-a)/sqrt(c*x^2 +
a)) + (8*A*c^2*x^4 + 25*B*a*c*x^3 + 16*A*a*c*x^2 + 10*B*a^2*x + 8*A*a^2)*sqrt(c*
x^2 + a)*sqrt(-a))/(sqrt(-a)*a*x^5)]

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Sympy [A]  time = 23.3522, size = 199, normalized size = 2.14 \[ - \frac{A a \sqrt{c} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{4}} - \frac{2 A c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 x^{2}} - \frac{A c^{\frac{5}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{5 a} - \frac{B a^{2}}{4 \sqrt{c} x^{5} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 B a \sqrt{c}}{8 x^{3} \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{B c^{\frac{3}{2}} \sqrt{\frac{a}{c x^{2}} + 1}}{2 x} - \frac{B c^{\frac{3}{2}}}{8 x \sqrt{\frac{a}{c x^{2}} + 1}} - \frac{3 B c^{2} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x} \right )}}{8 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+a)**(3/2)/x**6,x)

[Out]

-A*a*sqrt(c)*sqrt(a/(c*x**2) + 1)/(5*x**4) - 2*A*c**(3/2)*sqrt(a/(c*x**2) + 1)/(
5*x**2) - A*c**(5/2)*sqrt(a/(c*x**2) + 1)/(5*a) - B*a**2/(4*sqrt(c)*x**5*sqrt(a/
(c*x**2) + 1)) - 3*B*a*sqrt(c)/(8*x**3*sqrt(a/(c*x**2) + 1)) - B*c**(3/2)*sqrt(a
/(c*x**2) + 1)/(2*x) - B*c**(3/2)/(8*x*sqrt(a/(c*x**2) + 1)) - 3*B*c**2*asinh(sq
rt(a)/(sqrt(c)*x))/(8*sqrt(a))

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GIAC/XCAS [A]  time = 0.279818, size = 313, normalized size = 3.37 \[ \frac{3 \, B c^{2} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + a}}{\sqrt{-a}}\right )}{4 \, \sqrt{-a}} + \frac{25 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{9} B c^{2} + 40 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{8} A c^{\frac{5}{2}} - 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{7} B a c^{2} + 80 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{4} A a^{2} c^{\frac{5}{2}} + 10 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{3} B a^{3} c^{2} - 25 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )} B a^{4} c^{2} + 8 \, A a^{4} c^{\frac{5}{2}}}{20 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + a}\right )}^{2} - a\right )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + a)^(3/2)*(B*x + A)/x^6,x, algorithm="giac")

[Out]

3/4*B*c^2*arctan(-(sqrt(c)*x - sqrt(c*x^2 + a))/sqrt(-a))/sqrt(-a) + 1/20*(25*(s
qrt(c)*x - sqrt(c*x^2 + a))^9*B*c^2 + 40*(sqrt(c)*x - sqrt(c*x^2 + a))^8*A*c^(5/
2) - 10*(sqrt(c)*x - sqrt(c*x^2 + a))^7*B*a*c^2 + 80*(sqrt(c)*x - sqrt(c*x^2 + a
))^4*A*a^2*c^(5/2) + 10*(sqrt(c)*x - sqrt(c*x^2 + a))^3*B*a^3*c^2 - 25*(sqrt(c)*
x - sqrt(c*x^2 + a))*B*a^4*c^2 + 8*A*a^4*c^(5/2))/((sqrt(c)*x - sqrt(c*x^2 + a))
^2 - a)^5